Abstract-The digital Fourier transform (DFT) and the adaptive least mean square (LMS) algorithm have existed for some time. This paper establishes a connection between them. The result is the "LMS spectrum analyzer," a new means for the calculation of the DFT. The method uses a set of N periodic complex phasors whose frequencies are equally spaced from dc to the sampling frequency. The phasors are weighted and then are summed to generate a reconstructed signal. Weights are adapted to realize a best least squares fit between this reconstructed signal and the input signal whose spectrum is to be estimated. The magnitude squares of the weights correspond to the. power spectrum.For a proper choice of adaptation speed, the LMS spectrum analyzer will provide an exact N-sample DFf. New DFT outputs will be available in steady flow after the introduction of each new data sample. [l] have been known for a long time, and both of these techniques have enjoyed wide practical application. It is the purpose of this paper to demonstrate relationships between the DFT and the LMS algorithm by showing how the DFT can be calculated by making use of the LMS algorithm. This study leads to a new way to calculate the DFT. The resulting algorithm lends itself to parallel computation and VLSI implementation. Using a parallel implementation, processing speed is essentially independent of the size of the data block N. Computation time per data sample is thus also independent of the number of frequency points in the calculated spectrum. I. INTRODUCTION OTH THE DIGITAL Fourier transform (DFT) and the LMS adaptive algorithm
The high-luminosity LHC (HiLumi LHC) upgrade with planned operation from 2025 onward has a goal of achieving a tenfold increase in the number of recorded collisions thanks to a doubling of the intensity per bunch (2.2e11 protons) and a reduction of β Ã to 15 cm. Such an increase would significantly expedite new discoveries and exploration. To avoid detrimental effects from long-range beam-beam interactions, the half crossing angle must be increased to 295 microrad. Without bunch crabbing, this large crossing angle and small transverse beam size would result in a luminosity reduction factor of 0.3 (Piwinski angle). Therefore, crab cavities are an important component of the LHC upgrade, and will contribute strongly to achieving an increase in the number of recorded collisions. The proposed crab cavities are electromagnetic devices with a resonance in the radio frequency (rf) region of the spectrum (400.789 MHz). They cause a kick perpendicular to the direction of motion (transverse kick) to restore an effective head-on collision between the particle beams, thereby restoring the geometric factor to 0.8 [K. Oide and K. Yokoya, Phys. Rev. A 40, 315 (1989).]. Noise injected through the rf/low level rf (llrf) system could cause significant transverse emittance growth and limit luminosity lifetime. In this work, a theoretical relationship between the phase and amplitude rf noise spectrum and the transverse emittance growth rate is derived, for a hadron machine assuming zero synchrotron radiation damping and broadband rf noise, excluding infinitely narrow spectral lines. This derivation is for a single beam. Both amplitude and phase noise are investigated. The potential improvement in the presence of the transverse damper is also investigated.
a b s t r a c tThe Large Hadron Collider (LHC) relies on Landau damping for longitudinal stability. To avoid decreasing the stability margin at high energy, the longitudinal emittance must be continuously increased during the acceleration ramp. Longitudinal blowup provides the required emittance growth. The method was implemented through the summer of 2010. Band-limited RF phase-noise is injected in the main accelerating cavities during the whole ramp of about 11 min. Synchrotron frequencies change along the energy ramp, but the digitally created noise tracks the frequency change. The position of the noiseband, relative to the nominal synchrotron frequency, and the bandwidth of the spectrum are set by predefined constants, making the diffusion stop at the edges of the demanded distribution. The noise amplitude is controlled by feedback using the measurement of the average bunch length. This algorithm reproducibly achieves the programmed bunch length of about 1.2 ns 2 , at flat top with low bunch-tobunch scatter and provides a stable beam for physics coast. The noise can be injected either in the beam phase loop or directly in the cavity voltage set point. These two different technical implementations are presented and their respective advantages analyzed. The performance of the algorithm and its further applications are also presented in this paper.
Radio frequency (rf) accelerating system noise can have a detrimental impact on the Large Hadron Collider (LHC) performance through longitudinal motion and longitudinal emittance growth. A theoretical formalism has been developed to relate the beam and rf station dynamics with the bunch length growth [T. Mastorides et al., Phys. Rev. ST Accel. Beams 13, 102801 (2010)]. Measurements were conducted at LHC to determine the performance limiting rf components and validate the formalism through studies of the beam diffusion dependence on rf noise. As a result, a noise threshold was established for acceptable performance which provides the foundation for beam diffusion estimates for higher energies and intensities. Measurements were also conducted to determine the low level rf noise spectrum and its major contributions, as well as to validate models and simulations of this system.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.